Multi-objective optimization (MOO) or multi-criterion programming is one of the challenging problems encountered in various engineering problems such as e.g. the design of aerodynamic or hydrodynamic shapes. The present invention relates to the following multi-objective optimization problem (MOP) in continuous search space as shown in equation (1).min F(X)=(f1(X), . . . , fm(X))T; X∈Ω  (1)where X is the decision vector, F(X) is the corresponding objective vector, and Ω∈Rn is the decision space. Many evolutionary algorithms (EAs) have successfully been employed to tackle MOPs over the past decade. Several important techniques, such as the use of a second population (or an archive) have proved to be able to greatly improve the performance of EAs.
In contrast to single objective optimization, the distribution of the Pareto-optimal solutions often shows a high degree of regularity
The term “Pareto-optimal solutions” is well known in the field of MOO, see e.g. for this and other terms used in the present specification and the claims the glossary for terms in the field of Evolutionary Algorithms (EA) at http://ls11-www.cs.uni-dortmund.de/people/bever/EA-glossary/ which are incorporated by reference herein in its entirety.
Conventionally, this regularity has often been exploited implicitly by introducing a local search after evolutionary optimization. A step further to take advantage of such regularity is the use of a model that captures the regularity of the distribution of the Pareto-optimal solutions (Aimin Zhou, Qingfu Zhang, Yaochu Jin, Edward Tsang, and Tatsuya Okabe, A model-based evolutionary algorithm for bi-objective optimization, In Congress on Evolutionary Computation, Edinburg, U.K, September 2005, IEEE which is incorporated by reference herein in its entirety). In this paper, a linear or quadratic model is used in odd generations and a crossover and mutation in even generations to produce offspring.
The model-based offspring generation method used in the present invention is closely related to a large class of search algorithms known as estimation of distribution algorithms (EDAs) in the evolutionary computation community. EDAs first build probabilistic models to approximate the distribution of selected solutions in the population. Then, new solutions are generated by sampling from the probabilistic models. EDAs have been successfully used in single-objective optimization problems.
EDAs have also been extended for multi-objective optimization. In one known method, └τN┘ best performing solutions from the current population (N is population size and 0.0<τ<1.0) are selected first. Then the randomization Euclidean leader algorithm may be used to partition the selected points into several clusters. In each cluster, a Gaussian probability model is built to simulate the distribution of the solutions. Then N−└τN┘ solutions are sampled one by one from the models. This algorithm has been employed to solve both discrete and continuous problems.
Contrary to the conventional EDAs, the model in the multi-objective algorithm suggested by Aimin Zhou et al. (A model-based evolutionary algorithm for bi-objective optimization, cited above) consists of two parts, namely, a deterministic part and a stochastic part. The deterministic model aims to capture the regularity in the distribution of the population, while the stochastic model attempts to describe the local dynamics of the individuals. The model-based offspring generation method is then hybridized with the crossover and mutation in a heuristic way, i.e., in all odd generations the model-based method, and in all even generations the genetics-based method, is employed to generate offspring.